Transforming from cartesian to cylindrical and spherical

[yango_17]

Homework Statement

Translate the following equations from the given coordinate system into equations in each of the other two systems. Also, identify the surfaces so described by providing appropriate sketches.

Homework Equations

The Attempt at a Solution

For my solutions, I obtained z=2r^2 for the cylindrical equation and for the spherical equation I got:
(ρcosφ)^2 = 2(ρsinφcosθ)^2 + 2(ρsinφsinθ)^2. For my sketch I drew an infinite cylinder:

I was wondering whether my conversions were correct, as when I transform the same equation from cartesian to spherical and from cylindrical to spherical I seem to obtain different equations.

 

[Mark44]

Homework Statement

Translate the following equations from the given coordinate system into equations in each of the other two systems. Also, identify the surfaces so described by providing appropriate sketches.

Homework Equations

The Attempt at a Solution

For my solutions, I obtained z=2r^2 for the cylindrical equation and for the spherical equation I got:
(ρcosφ)^2 = 2(ρsinφcosθ)^2 + 2(ρsinφsinθ)^2. For my sketch I drew an infinite cylinder:

I was wondering whether my conversions were correct, as when I transform the same equation from cartesian to spherical and from cylindrical to spherical I seem to obtain different equations.

That's not a cylinder -- it's a cone. (Mathematically, cones have two parts.)

What was your equation in Cartesian form?

 

[yango_17]

Sorry I meant cone hahah. My original equation in cartesian form was z^2=2x^2+2y^2

 

[Mark44]

Sorry I meant cone hahah. My original equation in cartesian form was z^2=2x^2+2y^2

Then your cylindrical equation should be $z^2 = 2r^2$, not $z = 2r^2$ as you showed earlier.

 

[yango_17]

Does the spherical equation look correct?

 

[Mark44]

I'll take a look at it in a little while (it's dinner time...)

 

[yango_17]

Much appreciated (:

 

[Mark44]

(ρcosφ)^2 = 2(ρsinφcosθ)^2 + 2(ρsinφsinθ)^2

This is correct, as far as you went, but the right side could be simplified considerably.

 

[yango_17]

How would you go about simplifying it?

 

[Mark44]

How would you go about simplifying it?

Expand the terms on the right side.